Since \({M}_{B}\)  coincides with the left von Neumann algebra generated by the Hilbert algebra \({L}^{1,2}_{B}\) , then [1]} applies verbatim. Therefore, \(_B\)  coincides with the natural trace associated with the Hilbert algebra structure [1]}. As a consequence
\(_B\) 
is semi-finite, faithful and normal with ideal of definition \({L}^2_B\subset {M}_{B}\) . Unicity follows from [3]}.
